An integral domain is reduced and hence A is reduced

2. Integral Schemes

A scheme is integral if it is nonempty and for every nonempty affine open set U ⊂ X, the ring OX(U ) is an integral domain.

Proposition 2.1. A scheme is integral if and only if it is both reduced and irreducible.

Proof. Let X be an integral scheme. For every nonempty affine open U, the ring A = Γ(U, O_X) is an integral domain and U = Spec A. An integral domain is reduced and hence A is reduced. Since A is an integral domain, its nilradical Nil(A) = (0) is a prime ideal.

Hence U is irreducible. ¹ We find X is irreducible.

Conversely, assume that X is reduced and irreducible. Then for any nonempty affine open set U, A = O_X(U ) is reduced and U = Spec A is irreducible. Since A is reduced, Nil(A) = (0). Since U is irreducible, Nil(A) = (0) is a generic point of U. Hence (0) ∈ Spec A.

Then (0) is a prime ideal of A. By definition, A is an integral domain.



1An affine scheme is irreducible if and only it its nil radical is a prime ideal. See Reduced Schemes.

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